Anti-plane Analysis of a Circular Hole withThree Unequal Cracks in One-dimensionalHexagonal Piezoelectric Quasicrystals

This paper employes variable function method and the technique of conformal mappingto discuss the anti-plane problem of a circular hole with three unequal cracksin a one-dimensional （1D） hexagonal piezoelectric quasicrystal. Based on the piezoelectricityfundamental equations of quasicrystal materials and the symmetry of1D hexagonal quasicrystal and its linear piezoelectricity effect, 1D hexagonal quasicrystalcontrol equations of anti-plane problem are derived. Applying Cauchyintegral formula, the analytical expressions for the crack tip filed intensity factorsare presented with the assumption that the crack are electrical impermeable andelectrical permeable. With the variation of the hole-size and the crack length, someof the new model of crack are obtained. In the absence of the electric load, theresults match with the classical ones. The numerical results indicate the effects ofgeometric parameters on the field intensity factors. It is verified that the horizontalcrack length and the circle radius can easily promote crack growth. Researchon such issues will provide reliable theoretical value for the engineering materialspreparation and application.

Investigation on photonic crystal nanobeam cavity based on mixed diamond–circular holes

A photonic crystal nanobeam cavity(M-PCNC)with a structure incorporating a mixture of diamond-shaped and circular air holes is pro-posed.The performance of the cavity is simulated and studied theoretically.Using thefinite-difference time-domain method,the parameters of the M-PCNC,including cavity thickness and width,lattice constant,and radii and numbers of holes,are optimized,with the quality factor Q and mode volume Vm as performance indicators.Mutual modulation of the lattice constant and hole radius enable the proposed M-PCNC to realize outstanding performance.The optimized cavity possesses a high quality factor Q 1.45105 and an ultra-small mode=×volume Vm 0.01(λ/n)[Zeng et al.,Opt Lett 2023:48;3981–3984]in the telecommunications wavelength range.Light can be progres-=sively squeezed in both the propagation direction and the perpendicular in-plane direction by a series of interlocked anti-slots and slots in the diamond-shaped hole structure.Thereby,the energy can be confined within a small mode volume to achieve an ultra-high Q/Vm ratio.

Exact analytic solutions for an elliptic hole with asymmetric collinear cracks in a one-dimensional hexagonal quasi-crystal

Using the complex variable function method and the technique of conformal mapping, the anti-plane shear problem of an elliptic hole with asymmetric colfinear cracks in a one-dimensional hexagonal quasi-crystal is solved, and the exact analytic solutions of the stress intensity factors （SIFs） for mode Ⅲ problem are obtained. Under the limiting conditions, the present results reduce to the Griffith crack and many new results obtained as well, such as the circular hole with asymmetric collinear cracks, the elliptic hole with a straight crack, the mode T crack, the cross crack and so on. As far as the phonon field is concerned, these results, which play an important role in many practical and theoretical applications, are shown to be in good agreement with the classical results.

An analytical solution for the stress field and stress intensity factor in an infinite plane containing an elliptical hole with two unequal aligned cracks

The existing analytical solutions are extended to obtain the stress fields and the stress intensity factors(SIFs) of two unequal aligned cracks emanating from an elliptical hole in an infinite isotropic plane. A conformal mapping is proposed and combined with the complex variable method. Due to some difficulties in the calculation of the stress function, the mapping function is approximated and simplified via the applications of the series expansion. To validate the obtained solution, several examples are analyzed with the proposed method, the finite element method, etc. In addition, the effects of the lengths of the cracks and the ratio of the semi-axes of the elliptical hole(a/b) on the SIFs are studied. The results show that the present analytical solution is applicable to the SIFs for small cracks.

Strength and failure characteristics of sandstone containing two circular holes filled with two types of inclusions under uniaxial compression

Plate shaped sandstones containing two fabricated circular holes that were filled with gypsum and high-strength concrete respectively were prepared for studying the effects of ligament length L ligament incline angle α, as well as filling modes on their strength properties and failure modes. The results show that the initial cracks can be categorized as wing crack, axial tensile crack and curved tensile crack. The failure modes of ligaments can be categorized as mode of single inclined crack, mode of single axial crack and mode of two parallel cracks. The final failure modes of all specimens can be categorized as the tension-shear mixed failure and shear failure. The strength of inclusions shows little influence on the final failure modes of specimens, while the failure modes vary with L and α. When α is a fixed value, the peak strength σc and elastic modulus Ec of tested specimens increase firstly with increasing L and reaches to the maximum value at L of 16 mm, then declines. When L is a fixed value, σc declines firstly and then turns to increase as α increases to 75° from 45°, while Ec increases linearly. The axial stress σp performs the similar variation trends with those of σc versus increasing L and α when ligaments fail.

Analytic solutions to problem of elliptic hole with two straight cracks in one-dimensional hexagonal quasicrystals

By means of the complex variable function method and the technique of conformal mapping, the anti-plane shear problem of an elliptic hole with two straight cracks in one-dimensional hexagonal quasicrystals is investigated. The solution of the stress intensity factor （SIF） for mode III problem has been found. Under the condition of limitation, both the known results and the SIF solution at the crack tip of a circular hole with two straight cracks and cross crack in one-dimensional hexagonal quasicrystals can be obtained.

Screw dislocations interacting with two asymmetrical interfacial cracks emanating from an elliptical hole

The interaction between screw dislocations and two asymmetrical interfacial cracks emanating from an elliptic hole under loads at infinity is studied. The closed-form solution is derived for complex potentials. The stress intensity factor and the critical applied stress for the dislocation emission are also calculated. In the limiting cases, well-known results can be obtained from the present solutions. Moreover, new exact solutions for a screw dislocation interacting with some complicated cracks are derived. The results show that the shielding effect increases with the increase in the length of the other cracks and the minor semi axis, but decreases with the increase of dislocation azimuth. The repulsion acting on the dislocation from the other phase and the other crack extend in the horizontal direction, which makes the dislocation emission at the crack tip take place more easily, but the minor semi axis of the elliptical hole extending in the vertical direction makes it more difficult.

Interaction effects on heat conduction and thermal stress in an infinite elastic plane with two circular holes

Numerous researches have focused on the physical behavior of an elastic material in the vicinity of a single hole under the assumption that the interaction effects arising from the introduction of multiple holes remain negligible if the holes are placed sufficiently far from each other.In an effort to understand hole interaction effects on heat conduction and thermal stress,we consider the case when two circular holes are embedded in an infinite elastic material and use complex variable methods together with numerical analysis to obtain solutions describing temperature and elastic fields in the vicinity of the two circular holes.The results indicate that the interaction effects on temperature distribution and stress strongly depend on the relative size of the two holes and the distance placed between them but not on the actual size of the holes.

STRESS INTENSITY FACTORS OF A PLATE WITH TWO CRACKS EMANATING FROM AN ARBITRARY HOLE

In this paper, Muskhelishvili complex function theory and boundary collocation method are used to calculate the stress intensity factors (SIF) of a plate with two cracks emanating from an arbitrary hole. The calculated examples include a circular, elliptical, rectangular, or rhombic hole in a plate. The principle and procedure by the method is not only rather simple, but also has good accuracy. The SIF values calculated compare very favorably with the existing solutions. A t the same time,the method can be used far different finite plate with two cracks emanating from a hole with more complex geometrical and loading conditions. It is an effective unified approach for this kind of fracture problems.

Experimental Study and Numerical Simulation Using Extended Finite Element Method (XFEM) Combined with Cohesive Zone Model (CZM), of Crack Growth Induced by Non-Explosive Expansive Material on Two Neighboring Circular Holes of A Gneiss Rock

The Non-explosive expansion material (NEEM) is a method more environmentally friendly than the harmful conventional rock fracturing techniques. However, it is slower and very costly. Thus, any means of economizing their use is very desirable. This paper investigates the crack growth between two neighboring holes of a gneiss rock internally pressurized by NEEM mixed with water with the aim to evaluate the influence of holes spacing (center-to-center distance), on the initiation and growth of cracks. Field experimental results reveal that crack starts earlier and grows faster with increasing ambient temperature. But when the ambient temperature is above 28°C, the NEEM is “blown out” of the holes. At these ambient temperatures, the surrounding rocks are hot and cannot dissipate efficiently the heat generated by the hydration reaction. The best filling time was found to be in the evening when the daily hot temperature has drooped. The time to first crack increases as hole diameter decreases. The 3D numerical modeling and simulation of crack growth between two neighboring holes internally pressurized by NEEM using ABAQUS (XFEM/CZM) software shows a good agreement with the theoretical and experimental results.

Fracture evolution characteristics of sandstone containing double fissures and a single circular hole under uniaxial compression

The uniaxial compression experiments on the sandstone samples containing double fissures and a single circular hole were carried out by using electro-hydraulic servo universal testing machine to investigate the effect of rock bridge angle β and fissure angle α on mechanical properties and evolution characteristics of cracks.The results show that the peak strength,peak strain and elastic modulus of defected specimens decrease comparing with those for intact sample,and show a decreased trend firstly and then increase with β changing from 0° to 90°.The peak strength and elastic modulus achieve the minimum value as the rock bridge angle is 60°,while the peak strain reaches the minimum value with the rock bridge angle of 45°.The crack initiation of tested rock samples occurs firstly in stress concentration areas at tips of prefabricated fissures under uniaxial compression,and then propagates constantly and coalescences with the prefabricated hole.Some secondary cracks initiate and propagate as well until buckling failure happens.The rock bridge angle has a great influence on crack initiation,coalescence,final failure mode,crack initiation stress and transfixion stress.The peak strength varies significantly,while the elastic modulus and peak strain change slightly,and the failure modes are also different due to the influence of fissure angle.

Analytical Solutions of Cracks Emanating from an Elliptic Hole in an Infinite Plate under Tension

It is a common phenomenon that the cracks originating from a hole can cause structural damage in engineering.However,the fracture mechanics studies of hole edge crack problems are not sufficient.The problem of an elliptical hole with two collinear edge cracks of unequal length in an infinite plate under uniform tension at infinity is investigated.Based on the complex variable method,the analytical solutions of stress functions and stress intensity factors are provided.The stress distribution along the axes and the edge of the elliptical hole is given graphically.The numerical results show that there is obvious stress concentration near the hole and cracks,and the stresses tend to applied loads at distances far from the defect,which conform to Saint-Venant’s principle.Hence,the stress functions are proved to be right.Under special conditions,the present configuration becomes the Griffith crack,two symmetrical cracks emanating from an elliptical hole,two cracks of unequal length emanating from a circular hole,a crack at the edge of a circular hole,or a crack emanating from an elliptical hole.Compared with available results,stress intensity factors for these special shapes of ellipses and cracks show good coincidence.The stress intensity factor for two cracks of unequal length at the edge of an elliptical hole increases with the crack length and the major-to-minor axis ratio of the elliptical hole.The stress distribution in an infinite plate containing an elliptic hole with unsymmetrical cracks is given for the first time.

Exact Solutions for Piezoelectric Materials with an Elliptic Hole or a Crack under Uniform Internal Pressure

The existing investigations on piezoelectric materials containing an elliptic hole or a crack mainly focus on remote uniform tensile loads.In order to have a better understanding for the fracture behavior of piezoelectric materials under different loading conditions,theoretical and numerical solutions are presented for an elliptic hole or a crack in transversely isotropic piezoelectric materials subjected to uniform internal pressure and remote electro-mechanical loads.On the basis of the complex variable approach,analytical solutions of the elastic and electric fields inside and outside the defect are derived by satisfying permeable electric boundary condition at the surface of the elliptical hole.As an example of PZT-4 ceramics,numerical results of electro-elastic fields inside and outside the crack under various electric boundary conditions and electro-mechanical loads are given,and graphs of the electro-elastic fields in the vicinity of the crack tip are presented.The non-singular term is compared to the asymptotic one in the figures.It is shown that the dielectric constant of the air in the crack has no effect on the electric displacement component perpendicular to the crack,and the stresses in the piezoelectric material depend on the material properties and the mechanical loads on the crack surface and at infinity,but not on the electric loads at infinity.The figures obtained are strikingly similar to the available results.Unlike the existing work,the existence of electric fields inside an elliptic hole or a crack is considered,and the piezoelectric solid is subjected to complicated electro-mechanical loads.

ANISOTROPIC THERMOPIEZOELECTRIC SOLIDS WITH AN ELLIPTIC INCLUSION OR A HOLE UNDER UNIFORM HEAT FLOW

The two-dimensional problem of a thermopiezoelectric material containing an elliptic inclusion or a hole subjected to a remote uniform heat flow is studied. Based on the extended Lekhnitskii formulation for thermopiezoelectricity, conformal mapping and Laurent series expansion, the explicit and closed-form solutions are obtained both inside and outside the inclusion (or hole). For a hole problem, the exact electric boundary conditions on the hole surface are used. The results show that the electroelastic fields inside the inclusion or the electric field inside the hole are linear functions of the coordinates. When the elliptic hole degenerates into a slit crack, the electroelastic fields and the intensity factors are obtained. The effect of the heat how direction and the dielectric constant of air inside the crack on the thermal electroelastic fields are discussed. Comparison is made with two special cases of which the closed solutions exist and it is shown that our results are valid.

Displacement of surrounding rock in a deep circular hole considering double moduli and strength-stiffness degradation

The problem of cavity stability widely exists in deep underground engineering and energy exploitation.First,the stress field of the surrounding rock under the uniform stress field is deduced based on a post-peak strength drop model considering the rock’s characteristics of constant modulus and double moduli.Then,the orthogonal non-associative flow rule is used to establish the displacement of the surrounding rock under constant modulus and double moduli,respectively,considering the stiffness degradation and dilatancy effects in the plastic region and assuming that the elastic strain in the plastic region satisfies the elastic constitutive relationship.Finally,the evolution of the displacement in the surrounding rock is analyzed under the effects of the double modulus characteristics,the strength drop,the stiffness degradation,and the dilatancy.The results show that the displacement solutions of the surrounding rock under constant modulus and double moduli have a unified expression.The coefficients of the expression are related to the stress field of the original rock,the elastic constant of the surrounding rock,the strength parameters,and the dilatancy angle.The strength drop,the stiffness degradation,and the dilatancy effects all have effects on the displacement.The effects can be characterized by quantitative relationships.

THREE-DIMENSIONAL INTERACTIONS OF CIRCULAR CRACK IN TRANSVERSELY ISOTROPIC PIEZOELECTRIC SPACE WITH RESULTANT SOURCES

Exact solutions in form of elementary functions were derived for the stress and electric displacement intensity factors of a circular crack in a transversely isotropic piezoelectric space interacting with various stress and charge sources： force dipoles, electric dipoles, moments, force dilatation and rotation. The circular crack includes penny-shaped crack and external circular crack and the locations and orientations of these resultant sources with respect to the crack are arbitrary. Such stress and charge sources may model defects like vacancies, foreign particles, and dislocations. Numerical results are presented at last.

ELECTRIC FIELD GRADIENT EFFECTS IN ANTI-PLANE PROBLEMS OF A CIRCULAR CYLINDRICAL HOLE IN PIEZOELECTRIC MATERIALS OF 6 mm SYMMETRY

We study electromechanical felds in the anti-plane deformation of an infnite medium of piezoelectric materials of 6 mm symmetry with a circular cylindrical hole. The theory of electro- elastic dielectrics with electric feld gradient in the constitutive relations is used. Special attention is paid to the felds near the surface of the hole.

THE FUNDAMENTAL SOLUTIONS FOR THE PLANE PROBLEM IN PIEZOELECTRIC MEDIA WITH AN ELLIPTIC HOLE OR A CRACK

Based on the complex potential method, the Greed’s functions of the plane problem in transversely isotropic piezoelectric media with an elliptic hole are obtained in terms of exact electric boundary conditions at the rim of the hole. When foe elliptic hole degenerates into a crack, the fundamental solutions for the field intensity factors arc given. The general solutions for concentrated and distributed loads applied on the surface of the hole or crack are produced through the superposition of fundamental solutions With the aid of these solutions , some erroneous results provided previously in other works are pointed out More important is that these solutions can be used as the fundamental solutions of boundary element method to solve more practical problems in piezoelectric media.

Experimental research on mechanical properties of prestressed truss concrete composite beam encased with circular steel tube

Tests of 4 simply supported unbonded prestressed truss concrete composite beams encased with circular steel tube were carried out. It is found that the ratio of the stress increment of the unbonded tendon to that of the tensile steel tube is 0.252 during the using stage,and the average crack space of beams depends on the ratio of the sum of the bottom chord steel tube's outside diameter and the secondary bottom chord steel tube's section area to the effective tensile concrete area. The coefficient of uneven crack distribution is 1.68 and the formula for the calculation of crack width is established. Test results indicate that the ultimate stress increment of unbonded tendon in the beams decreases in linearity with the increase of the composite reinforcement index β0. The pure bending region of beams accords with the plane section assumption from loading to failure. The calculation formula of ultimate stress increment of the unbonded tendon and the method to calculate the bearing capacity of normal section of beams have been presented. Besides,the method to calculate the stiffness of this sort of beams is brought forward as well.

SCATTERING OF SH-WAVE BY CRACKS ORIGINATING AT AN ELLIPTIC HOLE AND DYNAMIC STRESS INTENSITY FACTOR

The method of complex function and the method of Green's function are used to investigate the problem of SH-wave scattering by radial cracks of any limited length along the radius originating at the boundary of an elliptical hole, and the solution of dynamic stress intensity factor at the crack tip was given. A Green's function was constructed for the problem, which is a basic solution of displacement field for an elastic half space containing a half elliptical gap impacted by anti-plane harmonic linear source force at any point of its horizontal boundary. With division of a crack technique, a series of integral equations can be established on the conditions of continuity and the solution of dynamic stress intensity factor can be obtained. The influence of an elliptical hole on the dynamic stress intensity factor at the crack tip was discussed.